Here is a detailed explanation for each sentence and the chosen option:
I'm told she is ill. I hope she will feel better again. ('A' is correct because 'will feel' is the appropriate future form to express hope for someone feeling better in the future.)
I was getting fired, so I stopped having lunch. ('B' is correct because 'having' is the gerund form used after the verb 'stop' to indicate an action that is no longer taking place.)
Do you walk to school every morning? ('B' is correct because 'do - walk' is the correct present simple question form for asking about regular actions.)
Hai enjoys going for a picnic at the weekend. ('B' is correct because 'going' is the gerund form used after 'enjoys' to express an activity someone likes.)
We needn't wear our coat. It won't be cold today. ('B' is correct because 'won't be' is the future form used for predicting something that will not happen.)
We like swimming because it's good for our health. ('A' is correct because 'like' is the simple present tense used for habits or preferences.)
My grandparents don't like living in the city because it's noisy and crowded. ('C' is correct because 'living' is the gerund form used after the verb 'don't like' to express a dislike for an activity or condition.)
Fire needs oxygen to burn. It doesn't burn without oxygen. ('D' is correct because 'doesn't burn' is the correct negative form in the present simple tense for a statement of fact.)
How much time do you spend reading book each day? ('C' is correct because 'reading' is the gerund form used after 'spend' to indicate the amount of time dedicated to an activity.)
I don't mind watching a DVD, but I prefer going to the cinema. ('D' is correct because 'watching' and 'going' are gerunds used after 'don't mind' and 'prefer', respectively.)